Subsection REF  displayed that the direct \(k\) -way \(p\) -spectral
clustering problem can be approximated as an optimization problem
on a Grassmann manifold.
Manifold optimization has been extensively developed over the last couple of decades, with the intention of providing robust numerical algorithms for problems on subspaces with a Riemannian structure. The work of [1]} and [2]} set the foundation to analyze such problems, with a focus on establishing a theory that leads to efficient numerical algorithms on the Stiefel \(\mathcal {St}(k,n)\)  and Grassmann \(\mathcal {Gr}(k,n)\)  manifolds. Specifically, they determine the Riemannian gradient and Hessian as the most critical ingredients in order to design first and second order algorithms on these subspaces.
In particular, the Riemannian gradient and Hessian are projections of their Euclidean counterparts onto the tangent space of the manifold
and the mapping between them is well established [1]}.
